We present expansions of real numbers in alternating s-adic series (1 < s ∈ N), in particular, s-adic Ostrogradskii series of the first and second kind. We study the “geometry” of this representation of numbers and solve metric and probability problems, including the problem of structure and metric-topological and fractal properties of the distribution of the random variable $$ {\xi } = \frac{1}{s^{{\tau_1} - 1}} + \sum\limits_{k = 2}^\infty {\frac{{\left( { - 1} \right)}^{k - 1}}{s^{{\tau_1} + {\tau_2} + ... + {\tau_k} - 1}},} $$ where τ k are independent random variables that take natural values.